Possible linguistic pedantry; he used "a" set of natural numbers, but "the" set of real numbers. It's equally difficult to represent the set of natural numbers.
To the question: I'm only saying that the "set of real numbers" and the "set of natural numbers" don't seem to exist.
Despite numerous downvotes, no one has yet produced them here (or even a link to them).
and here is one of the natural numbers: ℕ
Axiom of choice says I should be able to select an element from the set of real numbers (assuming it exists; but, I believe the assumption to be counterfactual).
I think it makes sense though; if I (claim to) have a thing, I should be able to choose/pick/select/point-to it (seems to be almost[?] tautological).
Both real and natural numbers can be reasoned about despite their infinite size (and we even know that e.g. there must be "more" real numbers than natural numbers, even though both sets are infinite)
There are no bounds in the universe that can contain all the real or natural numbers.